The structural and vibrational properties of two-dimensional hexagonal silicon (silicene) and germanium (germanene) are investigated by means of first-principles calculations. It is predict that the silicene (germanene) structure with a small buckling of 0.44 (0.7 ) and bond lengths of 2.28 (2.44 ) is energetically the most favorable, and it does not exhibit imaginary phonon mode. The calculated non-resonance Raman spectra of silicene is characterized by a main peak at about 575 cm(-1), namely the G-like peak. For germanene, the highest peak is at about 290 cm(-1). Extensive calculations on armchair silicene nanoribbons and armchair germanene nanoribbons are also performed, with and without hydrogenation of the edges. The studies reveal other Raman peaks mainly distributed at lower frequencies than the G-like peak which could be attributed to the defects at the edges of the ribbons, thus not present in the Raman spectra of non-defective silicene and germanene. Particularly the Raman peak corresponding to the D mode is found to be located at around 515 cm(-1) for silicene and 270 cm(-1) for germanene. The calculated G-like and the D peaks are likely the fingerprints of the Raman spectra of the low-buckled structures of silicene and germanene.